Generic 2-coverings of finite groups of lie type
Generic 2-coverings of finite groups of lie type
Summary: In [J. Algebra 59, 202-221 (1979; Zbl 0409.20033)] it was shown by \textit{R. H. Dye} that in a symplectic group \(G:=\text{Sp}_{2l}(2^f)=\text{Iso}(V,\langle\cdot,\cdot\rangle)\) defined over a finite field of characteristic 2 every element in \(G\) stabilizes a quadratic form of maximal or non-maximal Witt index inducing the bilinear form \(\langle\cdot,\cdot\rangle\). Thus \(G\) is the union of the two \(G\)-conjugacy classes of subgroups isomorphic to \(O^+_{2l}(2^f)\) and \(O^-_{2l}(2^f)\) embedded naturally. In this paper we classify all finite groups of Lie type \((G,F)\) with this generic 2-covering property (Theorem A). In particular, we show that there exists also an interesting example in characteristic 3, i.e., in the finite group of Lie type \(G:=F_4(3^f)\) every element in \(G\) is conjugate to an element of the subgroup \(B_4(3^f)\leq F_4(3^f)\) or of the subgroup \(3.{^3D_4(3^f)}\leq F_4(3^f)\).
- University of Florence Italy
- University of Udine Italy
- University of Milano-Bicocca Italy
Linear algebraic groups over finite fields, finite groups of Lie type, Simple groups: alternating groups and groups of Lie type, symplectic groups, covering properties, finite groups, maximal subgroups, Coverings; Finite groups of Lie type, conjugacy classes of subgroups
Linear algebraic groups over finite fields, finite groups of Lie type, Simple groups: alternating groups and groups of Lie type, symplectic groups, covering properties, finite groups, maximal subgroups, Coverings; Finite groups of Lie type, conjugacy classes of subgroups
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